Simplified Maximum Likelihood Detection for J£i Orthogonal Space-Time Block Coded System

i Adnan Ahmed Khan, 2 Khawaja TauseefTasneem, 2Syed Ismail Shah, Sr. MIEEE

iNational University of Sciences and Technology, Pakistan

2 Iqra University Islamabad Campus, H-9, Islamabad, Pakistan

1Emailofthecorrespondingauthor:adnankhan@mcs.edu.pk

Abstract- A Simplified Maximum Likelihood (ML) detector for Jf; Orthogonal Space-Time Block Coding (STBC) based on multi-step constellation reduction technique is presented. In the proposed algorithm, complexity is reduced by maintaining small ML search space, instead of searching over the entire constellation. The simulation results suggest that the proposed reduced search-space detection algorithm offers ML performance, while reducing the receiver complexity by more than 50% for the case of M-ary QAM and exactly 50% in case of M-ary PSK with M=16 as compared to the conventional ML detection. The complexity reduction is even more significant for higher modulation levels (M>16). The proposed technique is equally applicable to other STBC schemes.

I. INTRODUCTION

SPATIAL diversity is one of the well-known techniques

used to mitigate the adverse effects of a multipath

fading channel [1-3]. It is achieved by deploying multiple

antennas either at the transmitter or receiver, or at both the

sides. Depending on whether multiple antennas are used at

transmitter or receiver, space diversity can be classified into

receive diversity and transmit diversity. In the former case,

multiple antennas are used at the receiver side to pick up

independent copies of the transmitted signals whereas, in

the later, multiple antennas are deployed at the transmitter

side. In both the cases, multiple copies of the transmitted

signals are combined to mitigate the multipath fading

effects. Space-time block coding (STBC) is a transmit

diversity technique in which the data stream to be

transmitted is encoded in blocks, which are distributed

among multiple antennas and across time. Alamouti in [4]

proposed a simple STBC scheme for wireless

communication systems using two transmit antennas and a

linear maximum-likelihood (ML) decoder which was

generalized by Tarokh et. al. in [5-6] to an arbitrary number

of transmit antennas by applying the theory of orthogonal

designs. The main advantage of these schemes is a decoder

based on linear processing.

We introduce a constellation reduction technique based

decoding algorithm that simplifies the ML detection in

Orthogonal Space-Time Block Coded systems. The

proposed approach reduces the computational complexity of

these schemes while presenting ML performance.

Throughout this paper, we use the Jf4 STBC scheme

developed by Tarokh, but the idea presented here can easily be generalized to other STBC schemes.

Rest of the paper is organized as follows: System model is described in Section II. ML detector is discussed in Section III. Simplified ML detection technique is presented in Section IV. Simulations and results are discussed in Section-V. Algorithm complexity is discussed in section VI. Finally, Section VII concludes this paper.

II. SYSTEM MODEL

Consider an 1f..-encoded wireless communication system

employing four transmit antennas and one receiving

antenna. At each signaling interval, three complex symbols

SpS2 and s3are passed through an J{4 encoder which

transmits the symbols according to the following space-time

matrix

s, s, s,

s, ../2 ../2

-s, s, s, s, (I)

../2 -../2 C=[CifJ=

s, s, -s, -s;' +S2 -s; -S2 -s; +S, -s; ../2 ../2 2 2

s, S, S2 +s; +S. -s: s, +s; +S2 -s; ../2 -../2 2 2

where cij is transmitted by transmitter j at time i. Let the

channel gain from transmitter j to receiver at time i

be hj (i) . The received signal at time i is 4

r, = Ih/i)Cij + ni (2) j=l

Here, ni is zero mean, complex additive white Gaussian

noise (A WGN) with variance CJ'; . Channel is assumed to be

Rayleigh fading with zero mean and variance CJ'; :: I. Path

gain from transmitter j to receiver is hj , which is constant

for two symbol periods. Assuming perfect estimate of

Channel State Information (CSI) available at the receiver,

the manipulated received signal (MRS) vectors are defined

as follows [3, 8]

[ 'i+'i r,-r, . 'JT g,= J2 Tz r, r,

(4)

(5)

To obtain an expression of the channel matrix in relation

to the above MRS vectors, the real and imaginary parts of

the transmitted symbols are considered separately [8].

Let, Si = Xi + j Yi then s = [SI,S2,S3f = x+ j y (6)

where x=[X1,X2,X3]', y = [YPY"Y3]'

N ow using (I) - (6) the expression for the channel matrix in

relation to the MRS vectors is found to be [8]

gi = H~x+ j*H:y+vi i = 1, 2,3 (7)

where Vi is corresponding A WGN noise vector with zero

mean and variance 0'; . The channel matrices are

HR­,-

'" (I)

J.;(2)

h,(3)-h,(4)

J2 [11,(3) + h,(4)],

J2

h,(l)

h,(4)+h,(3)

J2 [h,(4)-h,(3)],

J2

h,(l)+ h,(I)

J2 [II, (2) - h, (2)],

J2 ",(3) - ",(4)+ h,(3) - h,(4)

2

[h,(3) + h,(3) + ",(4) - h,(4)],

2

2

II, (1) + h, (2) h, (1) - II, (2) 11, (1) + h. (I) + 11,(2) - h. (2)

J2 J2 J2 h,(I)-h,(2) h,(I)+h,(2) 11, (I) + h. (I) - 11, (2) + h. (2)

H~= J2 J2 J2

[11,(3) + h,(3)]' -11;(3) -h;(3)

2

-h;(4) h;(4) [II, (4)-h,(4)],

2

III. CONVENTIONAL ML DETECTOR

As we know that the conventional detector always

assumes the channel to be quasi-static. This implies constant

channel gains for 4'z:; i.e. hj(i) = hj' Substituting this

condition for quasi-static channel into (7) we get

i = 1, 2, 3 (8)

where the expressions for the perceived channel matrices

HR,H1can be found by substituting the assumed channel I I

condition into the H~ ,H: expressions above for i = 1, 2, 3.

An important property of the channel matrices H~, H: for a quasi-static channel given in [7] is "there exists a

solution to the following zero-forcing equation with respect

to 0i" (9)

48 distinct combinations in order to estimate the four

transmitted symbols.

The proposed technique is based on constellation

reduction. The method is quite general but for the sake of

simplicity we explain the idea for 16-QAMlPSK. After

applying orthogonal transformation H H on the received

signal vector r, we come up with the decision statistics

vector i, which forms the soft estimate of the transmitted

vector s.

At this stage, to find the estimate of each transmitted

symbol, conventional ML detector performs a search over

all the sixty four constellation points per symbol and decides

in favor of one that minimizes the criteria in (13) in order to

find the estimate of transmitted symbol.

Based on this observation, we present a simplified ML detector that uses a set of four reference constellation points

in four different quadrants as shown in fig. I and fig. 2.

K={a+ jb,-a+ jb,-a- jb,a- jb} for a,b>O (14)

The simplified ML estimate of a symbol is obtained in

two steps. In first step, ML criteria in (13) is applied to the

reference points in (14) (instead of the entire constellation

points) to reduce the search space for symbol under

detection to that particular quadrant whose reference point

minimizes the criteria in (13). During the second step, a

lookup table is first seen in order to get the other

constellation points associated with that reference

constellation point in (14) and then ML search in (13) is

once again performed with only these remaining points. The

symbol, which minimizes the ML criteria this time, is taken

as ML estimate of the transmitted symbol. Note that there is

no need to apply the ML criteria to the reference

constellation point the second time.

In case of 16-QAM % system, this method gives the

estimate of the four symbols by performing only 21 ML

searches. Similarly with 16-PSK modulation, this technique gives

the estimate of four transmitted symbols in 24 ML searches. This is due to the fact that two of the constellation points lie

exactly on quadrant boundaries, which can be addressed by

using rotated PSK constellation

The complexity reduction is even more significant for

higher modulation levels (greater than 16).

V. SIMULATIONS RESULTS

We assume four transmit and one receive antenna SML

constellation Reduction algorithm 16-QAM % system

under flat fading, quasi-static Rayleigh channel where the channel gains remain constant for three signaling intervals.

In fig.3 and fig.4 we used H4 system with 16-PSK and 16-QAM modulations respectively in a quasi-static Rayleigh fading channel. BER results show that the proposed Simplified ML detection possesses optimal BER

performance in both modulation schemes.

• Const PIs

I

2 ----~----+----~----~ .. '"

Rae Signal Ref PIs

1 1 0 Reduced PIs D SML Estimat I

I I I I

0 ML Estimate 1 ----~----.----~----~~~-.----~

I I

o - - - _.I- - - - - -I- - - - - -+ - - - - --< - - - - -1- - - --I I I ' I 1 I

I I I ~ I I

·1 - - - - ~ - - - - t - - - - ~ - - - -t- ----: ----~. ~ ____ L ____ ~ ____ ~ ____ ~ _____ I ___ _

-3

1 I 1

1

-2 -1

1 -,. Fig.! SML constellation Reduction algorithm 16-QAM

% system

I I " i ~ I '~; I I ... 0.8 - - t- - - t- - -"'t - -"'t - - --j - - -1- - -1- - -1- - -1- --

• I I I I f!) I 0.6 - - J,... __ .!.. __ .I.. __ ..l __ .....J ___ 1 ___ 1 ___ 1 ___ 1 __ _

I I I I

0.4 • - ~ - - ~ - -: • Constellation Pts 1:- - -:- - -:- ijf

0.2 - - I" - - T - - ~ ... Rae Signal pt i - - -:- - - :- - -

I I tJ Reference Pts I

()I - - t- - - -t- - - + (' ReducedSpace :- - -:- - -:- --,

-0.2 - - L - - ~ - -" D SML Estimate ,- - -1- - - 1- - -

I 0 ML Estimate I: I :

-0,4 • - ~ - - ~ - - : : - - -: - - - '- ---

I I I I I I I -0.6 - - - - - T - - T - - - - - -I - - - - - -1- - -1- - - 1- - -

• 1 1 1 • 1 -0.8 - - t- - - + - - + - - -;- - - -1 - - -I - - -1- - -1- - - 1- - -

• 1 • 1 1

Fig.2 SML constellation Reduction algorithm 16-

PSK % system

VI. COMPUTATIONAL COMPLEXITY

3

As we have already seen that, % space-time block coded

system employing 16-QAMIPSK modulation performs 16x3=48 searches in order to estimate four transmitted symbols if it uses a convention ML detector.

In contrast, for the same case of 16-ary modulations, the simplified ML detector first performs four searches to identify the quadrant in which the received symbol is located (which reduces the search space). In second step, it then further performs three searches in case of 16-QAM and four searches in case of 16-PSK to form the estimate of one symbol. Collectively, it performs only 7x3=21 searches for 16-QAM modulation and 8x3=24 searches for 16-PSK modulation in order to compute the estimates of four transmitted symbols. Note that this complexity reduction is even more significant for higher modulation levels.

Apart from low computational complexity, the proposed simplified ML detector gives us the same performance as the conventional ML as presented in [3].

a: w III

----~----+----~----~-----I-

____ L ____ 1 ____ J ____ ~ _____ I __ _

~~~~~~~~~!~~~~~~~~~§~~~~~I~~~-~

====~====~====~====~=====I=== = ----~----+----~----~-----I----

----r----T----'----~-----I----

10~~ ____ L_ ____ L_ ____ L_ ____ L_ _____ L_ __ ~

o 10 15 20 25 30 SNR

Fig.3: BER performance of proposed SML detection

with conventional % OSTBC ML 16-PSK

2 4 6 8 10 12 14 16 18 SNR

Fig.4: BER performance of proposed SML detection

with conventional ML in 16-QAM % OSTBC system

VII. CONCLUSION

A low complexity optimal ML detector for % encoded

system based on multi-step constellation reduction teclmique is presented. From simulation results, it is evident that the simplified ML detector performs analogous to the conventional ML detector, however it offers more than 50% reduction in computational complexity in case of QAM and exactly 50% reduction in computational complexity for PSK modulation. Using rotated PSK constellation in which the optimum rotation angle is selected such that there is no constellation point is on the quadrant boundary can further reduce the complexity in case of PSK signaling. Therefore, the proposed constellation reduction algorithm can be considered as a promising approach. This teclmique is equally applicable to the other STBC schemes like Alamouti

and q4 encoded systems.

4

REFERENCES

[I] G. J. Foschini, "Layered space-time architecture for wireless communication in a fading enviromnent when using multiple antennas," Bell Labs Tech. Journal, vol. I, pp. 41-59, Aut. 1996. [2] G. J. Foschini and M. J. Gans, "On limits of wireless communications in a fading environment when using mUltiple antennas," Wireless Personal Commun., vol. 6, pp. 311-335,1998. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless communications: performance criterion and code construction," IEEE Trans. Information Theory, vol. 44, pp. 744-765, Mar. 1998. [4] S. M. Alamouti, "A simple transmitter diversity scheme for wireless communications" IEEE J. Select. Areas Commun., vol. 16, pp. 1451-1458, Oct. 1998. [5] V. Tarokh, H. Jafarkhani, and A. R. Claderbank, "Space-time block codes from orthogonal designs," IEEE Trans. Inform. Theory., vol. 45, no. 5, July 1999. [6] V. Tarokh, H. Jafarkhani, and A. R. Claderbank, "Space-time block coding for wireless communications: Performance results," IEEEJ. Select. Areas Commun., vol. 17, no. 3, March 1999. [7] A. F. Naguib, N. Seshadri, and A. R. Claderbank, "Applications of Space-time block codes and interference suppression for high capacity and high data rate wireless systems," in Proc. 32nd

Asilomar Con! Signals, Systems, and Computers., Monterey CA, Nov. 1998,pp. 1803-1810 [8] F.-C. Zheng and A. G. Burr, "Signal detection for orthogonal space-time block coding over time-selective fading channels: The Hi systems," IEEE Trans. Wireless Commun., vol. 5, No.1, January 2006.